1. Is It Live, or Is It Fractal? Bergren Forum September 3, 2009 Addison Frey, Presenter

2. Chaos Under Control: The Art and Science of Complexity David Peak, Physics Department, Utah State University, Logan, UT 84322-4415 (PeakD@cc.usu.edu) and Michael Frame, Mathematics Department, Union College, Schenectady, NY 12308 (FrameM@union.edu)PeakD@cc.usu.eduFrameM@union.edu W.H. Freeman, Publishers 1994 ISBN 0-7167-2429-4

3. What Is a Fractal? Many fractals are “self-similar” (or nearly so). – But what is “self-similar”?

4. What Is a Fractal? Many fractals are “self-similar” (or nearly so). – But what is “self-similar”? A self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts).

5. What Is a Fractal? Many fractals are “self-similar” (or nearly so). – But what is “self-similar”? – Example:

6. What Is a Fractal? Many fractals are “self-similar” (or nearly so). – But what is “self-similar”? – Example:

7. What Is a Fractal? Many fractals are “self-similar” (or nearly so). – But what is “self-similar”? – Example: But a square is not a fractal!

8. What Is a Fractal? Many fractals are “self-similar” (or nearly so) A fractal has a fine structure at arbitrarily small scales.

9. What Is a Fractal? Many fractals are “self-similar” (or nearly so) A fractal has a fine structure at arbitrarily small scales. – Example:

10. What Is a Fractal? Many fractals are “self-similar” (or nearly so) A fractal has a fine structure at arbitrarily small scales. – Example:

11. What Is a Fractal? Many fractals are “self-similar” (or nearly so) A fractal has a fine structure at arbitrarily small scales. A fractal has a Hausdorff dimension greater than its Euclidean dimension

12. What Is Hausdorff Dimension? Example:

13. What Is Hausdorff Dimension? Example:

14. What Is Hausdorff Dimension? Example:

15. What Is Hausdorff Dimension? Example:

16. What Is Hausdorff Dimension? Example:

17. What Is Hausdorff Dimension? Example:

18. What Is Hausdorff Dimension? Example:

19. What Is Hausdorff Dimension? Example:

20. What Is Hausdorff Dimension? Example:

21. WARNING!

22. LOGARITHMS AHEAD!

28. What is the Hausdorff dimension of the Sierpinski carpet?

31. Fractals in Nature

34. Canacadea Creek, Alfred James Cahill

35. Process Box Counting Method – Place object in a single large box that leaves little to no extra space on the ends. – Apply a grid with smaller boxes over object, and count the number of boxes the object is in. – Repeat the second step with increasingly increasingly smaller boxes, and least twice more. James Cahill

36. The Math The number of boxes = N The scale factor (s) = Big box / Little box Plot graph with log(N) on the y-axis and log(s) on the x-axis. Create a best-fit line of the points. The slope of that line is the dimension of the object. James Cahill

37. N=1 S=1 N=17 S=11.02 N=38 S=20.84 James Cahill

38. N=116 S=60.8 James Cahill

39. The Creek is 1.166 Dimensional James Cahill

40. The box counting method is a slow, painstaking, but all together fairly accurate way to find the dimensions of natural objects. The idea behind it is that we take and average of the smaller and larger N values, and hope that it smoothes out any wrinkles in the results. Unlike mathematical fractals like the Sierpinski Gasket, our rivers and cracks lose definition at high magnification, so there comes a point when smaller S values are completely pointless, as the boxes are smaller than the thickness of the line we are examining. Concludatory James Cahill

41. What Is a Fractal? Many fractals are “self-similar” (or nearly so) It has a fine structure at arbitrarily small scales. It has a Hausdorff dimension greater than its Euclidean dimension It has a simple and recursive definition.

42. What Is a Fractal? Many fractals are “self-similar” (or nearly so) It has a fine structure at arbitrarily small scales. It has a Hausdorff dimension greater than its Euclidean dimension It has a simple and recursive definition. Example (A Deterministic Approach):

43. Generating the Sierpinski Gasket (Deterministic Approach)

52. Generating the Sierpinski Gasket (Random Approach) 0.Start with the point 1.Randomly choose one of the following: a. b. c. 2.Plot 3.Let 4.Go back to Step 1

53. Generating the Sierpinski Gasket (Random Approach)

67. A Maple Leaf

68. A Tree Jarrett Lingenfelter

69. A Tree Jarrett Lingenfelter

70. A Tree Jarrett Lingenfelter

71. A Tree Jarrett Lingenfelter

72. A Tree Jarrett Lingenfelter

73. A Tree Jarrett Lingenfelter

74. A Tree Jarrett Lingenfelter

75. Another Tree

78. WARNING!

79. TRIGONOMETRY AHEAD!

80. Another Tree (Random Approach) 0.Start with the point 1.Randomly choose one of the following: a. b. c. d. e. f. 2.Plot 3.Let 4.Go back to Step 1

81. Another Tree (Random Approach)

86. Another Tree (Second Attempt)

91. Another Tree (With Some Fruit)

92. A fractal landscape created by Professor Ken Musgrave (Copyright: Ken Musgrave).Ken Musgrave

93. A fractal planet.

94. Is It Live, or Is It Fractal?